Dan Colman of Open Culture recommends Star Trek Continues, "a critically-acclaimed, fan-produced webseries created by director and actor Vic Mignogna." I second his endorsement. The creators did an amazing job of capturing the fun and thrills of the original series.
If you ask the son of Gene Roddenberry, the creator of the original TV series, Star Trek Continues has managed to create a bona fide sequel. "I do have to say … I'm pretty damn sure my dad would consider this canon. The fact that you do stories that mean something, that have depth, that make us all think a little bit… I really think he would applaud you guys."
The Wall Street Journal adds to this:
[Star Trek Continues] comes frighteningly close to replicating the original series, in the sets, make-up and hairstyles, costumes and music… The art direction precisely captures the Day-Glo visuals of early color TV. Most remarkable is Mr. Mignogna; no actor playing, for instance, James Bond has imitated Sean Connery outright, but Mr. Mignogna comes so scarily close to the dynamic, staccato energy of William Shatner that we keep forgetting we're looking at another actor.
You can make a donation to support the 6th episode, due in May.